On transversally simple knots

Abstract

Final revision. To appear in the Journal of Differential Geometry. This paper studies knots that are transversal to the standard contact structure in $\reals^3$, bringing techniques from topological knot theory to bear on their transversal classification. We say that a transversal knot type $\cTK$ is {\it transversally simple} if it is determined by its topological knot type $\cK$ and its Bennequin number. The main theorem asserts that any $\cTK$ whose associated $\cK$ satisfies a condition that we call {\em exchange reducibility} is transversally simple. As a first application, we prove that the unlink is transversally simple, extending the main theorem in \cite{El}. As a second application we use a new theorem of Menasco (Theorem 1 of \cite{Me}) to extend a result of Etnyre \cite{Et} to prove that iterated torus knots are transversally simple. We also give a formula for their maximum Bennequin number. We show that the concept of exchange reducibility is the simplest of the constraints that one can place on $\cK$ in order to prove that any associated $\cTK$ is transversally simple. We also give examples of pairs of transversal knots that we conjecture are {\em not} transversally simple.